Scales to Play Over A#maj7
Every chord is a subset of several scales: any scale that contains all of a chord's tones fits over it. Pick a chord to see the matching scales from its own root, the major keys the chord belongs to, and full fretboard diagrams for every scale on guitar.
An A#maj7 chord contains the notes A# – C## – E# – G##. Any scale that includes every one of those 4 tones fits over it — from the root A#, that means A# Major, A# Lydian and A# Mixolydian Bebop. They differ only in the notes between the chord tones, and that is where each scale gets its color.
A#maj7 is also diatonic to 2 major keys: Bb major (relative minor Gm), F major (relative minor Dm). In a song from one of those keys you can also solo with the key's own major scale or its relative minor pentatonic across the whole progression, not just this chord.